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String analysis is the problem of reasoning about how strings are manipulated by a program. It has numerous applications including automatic detection of cross-site scripting (XSS). A popular string analysis technique includes symbolic executions, which at their core use string (constraint) solvers. Such solvers typically reason about constraints expressed in theories over strings with the concatenation operator as an atomic constraint. In recent years, researchers started to recognise the importance of incorporating the replace-all operator and finite transductions in the theories of strings with concatenation. Such string operations are typically crucial for reasoning about XSS vulnerabilities in web applications, especially for modelling sanitisation functions and implicit browser transductions (e.g. innerHTML). In this paper, we provide the first string solver that can reason about constraints involving both concatenation and finite transductions. Moreover, it has a completeness and termination guarantee for several important fragments (e.g. straight-line fragment). The main challenge addressed in the paper is the prohibitive worst-case complexity of the theory. To this end, we propose a method that exploits succinct alternating finite automata as concise symbolic representations of string constraints. Alternation offers not only exponential savings in space when representing Boolean combinations of transducers, but also a possibility of succinct representation of otherwise costly combinations of transducers and concatenation. Reasoning about the emptiness of the AFA language requires a state-space exploration in an exponential-sized graph, for which we use model checking algorithms (e.g. IC3). We have implemented our algorithm and demonstrated its efficacy on benchmarks that are derived from XSS and other examples in the literature.